Are there other symmetries on the plane besides the wallpaper and point groups?

[mnb96]

Hello,

is it so that symmetries on the plane are essentially discrete subgroups of the group of isometries on the plane?
If that is true, then why should we think that the only symmetries in the plane are given by the wallpaper group and the point group? Can't we just change the metric of the 2D plane and obtain new kind of symmetries?

For instance, could we just consider the space ℝ^1,1 with norm ||p||²=x²-y², where p=(x,y), and find new subgroups of isometries w.r.t. this metric?

 

[HallsofIvy]

One of the reasons symmetries are important is that they are independent of the metric. Defining a new metric does not define new symmetries.

 

[mnb96]

One of the reasons symmetries are important is that they are independent of the metric.

Then there is something that I misunderstood.

For instance, I could make a symmetry in the plane in the following way: map the 2D Euclidean plane (minus the origin) into the log-polar domain, then define a translational symmetry across the "radial" coordinate, and apply the inverse map back to Euclidean space: wouldn't this calculation produce a sort of "scaling symmetry" in the plane?

Of course, that would not be an isometry w.r.t. the Euclidean metric, but rather an isometry in log-polar domain. Or am I wrong?

 

[micromass]

It all depends on what you mean with a symmetry. A symmetry is only defined by an underlying group. So when somebody talks to me about symmetries of the plane, then I automatically think of the Euclidean metric-preserving functions. That is how I would define a symmetry of the plane. This leads to the group $O(2)$ (+ translations). If you want to talk about functions preserving some other metric, then you can of course do that, but those symmetries will have a different status than then $O(2)$-symmetries. For example, you could have the $O(1,1)$-symmetries.

 

[mnb96]

Thanks Micromass for your answer.

If you say that one could, in principle, define symmetries preserving some other metric, then why HallsofIvy was suggesting that symmetries are "independent of the metric"? Could we clarify this point?

Furthermore, some time ago I came across some paper talking about some "fancier" symmetries involving hyperbolic geometry that can be visually appreciated in some of Escher's drawings. Are also these symmetries represented by groups of transformations that preserve some other (non-Euclidean) metric or not?

 

[micromass]

If you say that one could, in principle, define symmetries preserving some other metric, then why HallsofIvy was suggesting that symmetries are "independent of the metric"? Could we clarify this point?

I don't know what HallsOfIvy is talking about. In my point of view, symmetries depend on the metric.

Furthermore, some time ago I came across some paper talking about some "fancier" symmetries involving hyperbolic geometry that cab be visually appreciated in some of Escher's drawings. Are also these symmetries represented by groups of transformations that preserve some other (non-Euclidean) metric or not?

Yes. The hyperbolic plane can be seen as a regular disk (or some other space) equipped with some non-Euclidean metric.

 

[mnb96]

Ok. So, is it correct to say that "symmetries" are essentially "groups of isometries w.r.t. some given metric"?

 

[micromass]

That is one way to see symmetries. This is the point of view of transformation geometry.

 

[mnb96]

Alright. Thanks a lot! This is what I wanted to know.

In the meanwhile I am reading a paper about symmetries on the hyperbolic plane. It says that such symmetries can be seen as "isometries of the hyperbolic plane", and that "there are four basic isometries in the hyperbolic plane: non-Euclidean reflection, non-Euclidean rotation, non-Euclidean translation, and parabolic isometry". Very interesting.